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  <h1 id="线性代数" class="content-subhead">线性代数</h1>
  <p>
    <span>1970-01-01</span>
    <span><span class="post-category post-category-math">Math</span></span>
    <span id="/public/article/线性代数.html" class="leancloud_visitors" style="display:none" data-flag-title="线性代数"></span>
  </p>
  <h4 id="_1">三角矩阵</h4>
<p>上三角矩阵<br />
<script type="math/tex; mode=display">
A=
\left[\begin{array}{c c c}
a_{11}&a_{12}&\cdots&a_{13} \\
      &a_{22}&\cdots&a_{23} \\
      &      &\ddots&\vdots \\
      &      &      &a_{nn} 
\end{array}\right]
</script>
<br />
下三角矩阵<br />
<script type="math/tex; mode=display">
A=
\left[\begin{array}{c c c}
a_{11}&      &      &       \\
a_{21}&a_{22}&      &       \\
\vdots&\vdots&\ddots& \\
a_{n1}&a_{n2}&\cdots&a_{nn} 
\end{array}\right]
</script>
<br />
三对角矩阵<br />
<script type="math/tex; mode=display">
A=
\left[\begin{array}{c c c}
a_{11}&a_{12}&      &       \\
a_{21}&a_{22}&\ddots&       \\
      &\ddots&\ddots&a_{n-1,n} \\
      &      &a_{n,n-1}&a_{nn} 
\end{array}\right]
</script>
</p>
<h4 id="_2">对角矩阵</h4>
<p>
<script type="math/tex; mode=display">
A=
\left[\begin{array}{c c c}
a_{11}&      &      &       \\
      &a_{22}&      &       \\
      &      &\ddots&       \\
      &      &      &a_{nn} 
\end{array}\right]
</script>
</p>
<h4 id="_3">对称矩阵</h4>
<p>
<script type="math/tex; mode=display">
A = A^T
</script>
</p>
<h4 id="_4">埃尔米矩阵</h4>
<p>矩阵等于它自己的共轭转置矩阵<br />
<script type="math/tex; mode=display">
A = A^*
</script>
</p>
<h4 id="_5">正定矩阵</h4>
<p>若 <script type="math/tex">A = A^T</script>，对任意非0向量 <script type="math/tex">x</script>，满足 <script type="math/tex">Q(x)>0</script>，则 <script type="math/tex">A</script> 为正定矩阵<br />
<script type="math/tex; mode=display">
Q(x) = x^TAx
</script>
<br />
半正定矩阵：同上， <script type="math/tex">Q(x) \geq 0</script>，则A为半正定矩阵</p>
<p>半负定矩阵：同上， <script type="math/tex">Q(x) \leq 0</script>，则A为半负定矩阵</p>
<p>不定矩阵：若A既不是半正定矩阵，也不是半负定矩阵，则A为不定矩阵</p>
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  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#_1">三角矩阵</a>
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  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#_2">对角矩阵</a>
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  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#_5">正定矩阵</a>
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